26:3 FOURTH REPORT — 1834. 



The attraction of the sun-ounding fluid on the points of the 

 canal so situated at the extremity which terminates in the hori- 

 zontal surface, will produce a pressure, which Laplace calls K, 

 on the canal doivmvards. If a tangent plane be drawn at the 

 other extremity, the fluid below this plane will produce an equal 

 pressure on the canal at this end, and similarly directed. These 

 two pressures acting in opposite directions along the canal, will 

 destroy each other by reason of the incompressibility of the 

 fluid. There will remain the attraction of the fluid between the 

 curve surface and the tangent plane. This produces a pressure 

 directed to the centres of curvature of the point where the canal 

 ends, and, as the calculation shows, proportional to the sum of 

 the greatest and least curvatures at that point ; for, in fact, the 

 quantity of matter between the curve surface and tangent plane, 

 taken within the small distance X from the point of contact, 

 varies in the same proportion, and to this quantity of matter 

 the total attracting force is proportional. Opposed to the pres- 

 sure thus arising is the effect of gravity on the whole canal in 

 producing pressure in the direction of its length, which effect, 

 it is known from the common principles of hydrostatics, is 

 equal to the weight of a column of the fluid of the same trans- 

 verse section as the canal, and whose height is the elevation of 

 one end of the canal above the horizontal plane in which the 

 other is situated. Calling this elevation 2, the greatest and least 

 radii of curvature at the point of the curved surface under con- 

 sideration R and R', and the density of the fluid g, we shall have 



H / 1 1\ 



y Vr + R7;=^^^- 



This is the fundamental equation spoken of above. It does 

 not contain, as we perceive, the quantity K, which Laplace sup- 

 poses to be expressive of the force that causes the suspension 

 before mentioned (p. 256) of mercury in the tube of a barometer 

 to a height two or three times greater than that due to the at- 

 mospheric pressure. He thinks also that on this quantity de- 

 pends the forces which produce cohesion and chemical affinities. 

 The left side of the equation, expressing the pressure that arises 

 from the action of the small quantity of matter situated between 

 the curve surface and the tangent plane, and circumscribed by the 

 surface of the sphere of activity whose centre is the point of con- 

 tact and radius X, must be exceedingly small compared to K. 



The above equation cannot be generally integrated ; but in 

 the case in which it belongs to a surface of revolution the axis 

 of which is vertical, as, for instance, when the capillary tube is 

 cylindrical with a circular base, an integral is obtained which 



